On The Observational Implications of Taste-Based Discrimination in Racial

Profiling

William A. Brock, Jane Cooley, Steven N. Durlauf and Salvador Navarro

February 11, 2010

Revised December 20, 2010

Department of Economics, University of Wisconsin at Madison. This work was supported by the National Science Foundation grant SES-0518274 and the University of Wisconsin Vilas Trust, University of Wisconsin Graduate School, Institute for Research on Poverty, and Laurits Christensen Chair in Economics. We thank Hon Ho Kwok, Xiangrong Yu and Yu Zhu for excellent research assistance. We thank Elie Tamer and an anonymous referee for very helpful suggestions. Daniel Nagin, Alexis Piquero, David Weisburd, Sarit Weisburd, and especially Angela Duckworth have helped us with the criminological and psychological literatures. This paper was prepared for the festschrift volume in honor of Charles Manski and is written in his honor.

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1. Introduction

Minorities are generally subject to higher rates of policing, such as automobile

or pedestrian stops, than whites. For instance, Ridgeway (2007) reports that 87 percent of

stops of the New York police department in 2006 were nonwhite, and 51 percent of these

were black. These differentials are often referred to as racial profiling. Racial profiling

has generated and continues to generate an active scholarly literature because differential

treatment by police officers (often measured as stop or search rates) can stem from a

range of factors, not all of which correspond to what is commonly understood as

discrimination. A central difficulty in the study of racial bias stems from the fact that the

police officer is likely to have more information about the potential guilt of suspects than

the analyst. This makes it difficult to attribute racial disparities in police treatment of

potential criminals to discrimination. From a policy perspective, this distinction is likely

to be very important, as racial disparities that result from optimal (or productive) policing

may be less of a concern than disparities that result from discrimination. In this paper,

we examine the challenges to identifying discrimination in police behavior in

observational data.

For our purposes, we equate discrimination with Becker’s (1957) notion of

taste-based discrimination (or racial prejudice). As originally argued in Knowles, Persico

and Todd (2001), which we subsequently denote as KPT, and recently reviewed in

Persico (2009) differential treatment of blacks and whites does not imply the presence of

taste-based discrimination. The objective functions of police could be race neutral yet

produce differential equilibrium search rates by race.1

1This latter differential may be attributed to statistical discrimination. Our focus on taste-based discrimination does not imply a justification for statistical discrimination; in fact the ethics of statistical discrimination have been challenged by Harcourt (2004) and Sklansky (2008); see Risse and Zeckhauser (2004) for a defense. However, all of these positions recognize an ethical distinction between taste-based discrimination and statistical discrimination and hence the importance of empirically distinguishing them.

This has spurred a literature on the

potential to detect taste-based discrimination using outcome data (such as arrest rates), a

prominent example of which is a recent Rand study (Ridgeway, 2007), of pedestrian

stops in New York City.

2

Our goal in this paper is to describe conditions under which taste-based

discrimination in police stops and searches can be uncovered in observational data. One

important contribution relative to the existing literature is that we consider the robustness

of various tests for discrimination to heterogeneity across groups of potential criminals

and officers.

With respect to potential criminals, we are concerned with the possibility that

police stops involve information that is not available to the econometrician. KPT is also

concerned with heterogeneity across potential criminals that may be observed to police

officers but not the econometrician. If the payoffs to carrying contraband vary with

characteristics correlated with race, such as income, then we might observe higher search

rates for nonwhites even in the absence of taste-based discrimination. However, KPT

shows that when police use this information to help maximize search success rates in

their context, individuals with the “guilty characteristic” have the incentive to carry

contraband less and those without the characteristic to carry contraband more. In

equilibrium, the probability of carrying contraband equalizes across groups and so the

heterogeneity provides no useful information with which to identify criminals.

Differentials in the search success rates across race only occur in their model when police

discriminate. We show that in a more general model where the police officer’s utility

depends on the intensity of search rates, the problem of unobservables resurfaces because

search success rates no longer equalize across groups in the absence of discrimination.

Like us, Anwar and Fang (2006) (henceforth denoted as AF) and Bjerk (2007)

are also concerned with unobservables that could prevent attributing differential success

rates across racial groups to taste-based discrimination. However, they focus on a

particular type of unobservable that signals the guilt of the potential criminal to the

officer. Our analysis permits unobservables to play a more general role in that we do not

restrict them to just act as a signal to the officer. In addition, unobservables can affect

equilibrium guilt probabilities and enter into officers’ preferences. Thus, their whole

distribution matters for detecting bias, as highlighted in Heckman (1998).

Beyond heterogeneity in potential criminals, we also contribute to the literature

by highlighting the role of heterogeneity across individual police officers in taste for

discrimination. In our view, this heterogeneity is essential for capturing Becker’s original

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thinking, in which the key determinant of degrees of discrimination involves the behavior

of the marginal employer in a labor market. In particular, we consider the possibility that

officers have differential tastes both for searching guilty criminals and for searching

potential criminals regardless of guilt.

AF is also concerned with police officer heterogeneity, but a particular type of

heterogeneity where the cost to searching criminals of different races varies by the race of

the officer. They exploit this heterogeneity to derive testable implications about relative

racial prejudice using a ranking condition that compares searches and success rates of

criminals of different races across officers of different races. We discuss how our

formulation links to AF’s and argue that their rank condition is sensitive to model

assumptions in a fashion similar to KPT. We further illustrate a set of assumptions

through which we can exploit police officer heterogeneity to identify relative bias in our

model.2

The literature notes reasons beyond unobservables and police officer

heterogeneity why a test based on search success rates may fail. Dominitz and Knowles

(2006) show that the condition of equal search success rates, while consistent with KPT’s

model of arrest rate maximization, would generally not hold in a model where the police

officer objective was to minimize crime. Thus, like us, they are concerned with the

sensitivity of the result of equal search success rates to the specification of the police

officer objective function. However, we consider the opposite possibility, that equal

search success rates can be consistent with crime minimization. We show that equal

search success rates neither necessarily distinguish between arrest maximization and

crime minimization as objective functions nor rule out the possibility of racial prejudice.

In terms of substantive conclusions, we make four claims. First, tests based on

the cross groups comparisons of either the success rate of searches or search rates are not

informative about the presence or absence of taste-based discrimination when one

weakens the assumptions made by KPT and Persico (2009). Second, the presence or

2Antonovics and Knight (2009) also consider the potential for exploiting police officer heterogeneity as a means of identifying taste-based discrimination. Unlike AF, they focus on search rates (rather than search success rates) in a context where like KPT the unobservable (to the econometrician) does not act as a signal of guilt.

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absence of taste-based discrimination does not necessarily differentiate between crime

minimization and a decentralized determination of stop and search strategies except for

relatively special assumptions. Third, even if a police department possesses superior

information to an econometrician, it is possible for discrimination to go undetected when

there is unobserved taste heterogeneity across police officers. A policy mandating

officers achieve either equal search or success rates across observable groups would not

prevent biased officers from exercising their bias. Fourth, by placing restrictions on

unobserved heterogeneity it is possible to develop testable implications for detecting bias

even in our more general setting. Together, these results suggest that model-specific

claims about taste-based discrimination and racial profiling should be interpreted with

caution; in our conclusions we discuss constructive ways to proceed given this.

It is worth observing that racial profiling studies require a somewhat different

view of the process by which taste-based discrimination affects outcomes than Becker’s

original model. In Becker’s analysis, which focused on labor markets, equilibrium wage

gaps between blacks and whites is determined by the behavior of the marginal

discriminator in a labor market; see Charles and Guryan (2008) for a recent test of this

proposition. From this vantage point, it is possible for a market to contain prejudiced

firms even though the wage gap between blacks and whites is zero; this may happen

when potential discriminators employ all white workforces. An analogous argument does

not directly apply to the study of police stops and searches. Each potential discriminator

separately interacts with a common population and, ceteris paribus, would always prefer

to sample blacks if the option arises. A segregating firm with an all white workforce has

no incentive, at equal wage rates, to substitute a black worker for a white one.

Section 2 of this paper outlines a general model of police stops and equilibrium

guilt rates. We use this abstract vantage point to argue that tests of taste-based

discrimination based on simple criteria such as equality of hit rates or search rates are not

informative about taste-based discrimination without very particular assumptions on

preference structures3

3We do not claim a general observational equivalence result nor do we argue that field experiments would not potentially be able to reveal taste-based discrimination in racial profiling.

. The need for such assumptions does not render previous exercises

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uninteresting but rather indicates the importance of performing sensitivity analyses.

Section 3 studies a generalization of the KPT model, one that follows aspects of its

formulation in Persico (2009) focusing on predictions deriving from search success rates.

Here we discuss AF (2006) in some detail. Section 4 considers how biased officers can

survive police department policies to detect bias in the presence of unobservable officer

heterogeneity. Section 5 considers empirical predictions of the absence of taste-based

discrimination in our model using assumptions on unobservables. We show how

different restrictions can lead to both point and partial identification results. Section 6

concludes.

Section 2. Basic model

i. decision problems for potential criminals and police officers

We consider the behavior of two populations: a population of civilians who make

choices as to whether or not to engage in an illegal activity (for concreteness, we think of

the activity as carrying contraband) and a population of police officers each of whom

chooses a strategy for stopping individuals who will be searched.

Each member 1,...,j N= of the civilian population is a member of some group.

The characteristics of a group are described as a triple, ( ), ,o ur c c , where r denotes the

race of the group members, oc denotes the characteristics of individuals in the group that

are observable to both the econometrician and the officer, and uc denotes the

characteristics of individuals in the group that are observable to the officer but not to the

econometrician. The number of groups is assumed to be finite. We will denote ,, o ur c cn as

the number of members of a group. For example, an officer may follow a stop strategy

that depends on a driver’s race, car type and driving style, the last of which is unavailable

6

to the analyst.4

We employ groups to define what it means for an officer to regard two individuals

as indistinguishable. We assume that individual heterogeneity is not observable to the

police, so group level characteristics summarize what police officers know about an

individual. This is without loss of generality as one can always define a finer partition of

individuals across groups in which the observable part of the individual heterogeneity is

treated as a group attribute. The primary identification challenge is the presence of the

characteristics that the police officer observes that the researcher does not ( as opposed

to ).

In addition, we assume that individual specific heterogeneity among

members of a group is described by , which is taken as independent across individuals.

For individual j , a choice is made between committing a crime, for the profiling

case this typically amounts to the possession of contraband, C , and not committing a

crime, i.e. not carrying contraband, NC . We assume that all uncertainty associated with

the payoff to this choice involves the possibility of being searched or not being searched

( S or NS ). Therefore, the individual’s expected utility from carrying

contraband/commission of a crime equals

( ) ( ) ( ) ( ), , , , , , , ,, , , , , , , 1 , , , ,o u o u o uC o u r c c j r c c C S o u j r c c C NS o u jEU r c c s s U r c c s U r c cε ε ε= + − (1)

where , ,o ur c cs is the probability of being searched if one is a member of the group. As

noted above, jε is not observed by an officer, hence an individual uses the probability of

a member of his group being searched to assess his own probability of being searched.

We assume that civilians have rational expectations in the sense that their subjective

probabilities concerning searches will correspond to the equilibrium search probabilities

of the complete model.

4Grogger and Ridgeway (2006) argues that the race of a driver is not known at night if one is focusing on car stops; their approach means group definitions are time dependent as some variables only periodically appear in the officer’s information set; we do not pursue this generalization of our information structure.

7

Normalizing the utility of no crime commission to zero, an individual will carry

contraband if equation (1) is positive. From the perspective of a given police officer,

since jε is unobservable, the solutions of the individual-level crime choice problems will

produce group-specific crime rates

( )( ), , , , , ,Pr , , , , 0 , , , ,o u o u o ur c c C o u r c c j o u r c cEU r c c s r c c sπ ε= > (2)

which summarize the information available to the police on relative guilt probabilities

between individuals. Police are assumed to have rational expectations in the sense that

they employ these probabilities in their search calculations.

The crime choices made by civilians are paralleled in the search choices made by

the police. Officer 1,...,i M= chooses group-specific search rates , , ,o ui r c cs . We are not

explicit here about the constraints that search rates must satisfy. In most of our analysis,

search rates will be have to be consistent with a fixed number of stops per day; we will

consider additional restrictions placed by the police department. We allow for

heterogeneity in search rates across officers. We distinguish between officer types with

observable characteristics ,o ip and unobservable characteristics ,u ip with respect to what

is observable to the econometrician.

The expected utility for a police officer with characteristics ,o ip , ,u ip of applying a

search rate ,, o ur c cs to a group characterized by ( ), ,o ur c c is

( ) ( ) ( ), , , , , , , , , , , , , ,, , , , , 1 , , , , , .o u o u o u o ur c c C o i u i u i r c c r c c NC o i u i oo u i r c cV p p r c c s V p p r c c sπ π+ − (3)

Here ( ), , , , ,, , , , ,o uoC o i u i u i r c cV p p r c c s denotes the utility of searching a guilty member of the

group and ( ), , , , ,, , , , ,o uNC o i u i u i r co cV p p r c c s denotes the utility of searching an innocent

member. We include the individual search rate , , ,o ui r c cs in the utility function as it allows

for the possibility that the utility to a given search depends on the intensity of search the

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officer applies to the group. Notice the presence of this term does not imply taste-based

discrimination per se. For example, it may be interpreted as a cost parameter in the sense

that different choices of search allocations across groups involve the use of different

locations at which to search civilians.

Let denote the number of members of a group searched by officer , i.e.

. Aggregating across police determines the probability of being

searched via

,,

, ,, , ,

,,

,

.o u

o u o u

o u

i r cr c i r c

r c

cc c

c

a dis s di

n= = ∫∫ (4)

Combining (4) with (2), guilt probabilities for each group can be thought of as a function

( ),, , ,, , ,o u o ur c o rc u ccr c scπ π= . (5)

This closes the description of the crime choices of civilians and the search choices of

police officers, once one is explicit about the restrictions on search rates that an officer

must fulfill.

We do not prove the existence of equilibrium for the joint crime and search

choices of civilians and police. Establishing sufficient regularity or smoothness

conditions on preferences and search constraints in order to ensure a Nash or a

Stackelberg equilibrium is not relevant to our goals in this paper. We will prove

existence for the version of this model we study in the next section, so existence is by no

means vacuous.

ii. defining taste-based discrimination

Our formulation of the individual officer decision problems provides one natural

definition for the absence of discrimination in the preferences of a given police officer,

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namely irrelevance of race in an officer’s utility function given other factors. Formally,

the absence of a taste for race-based discrimination on the part of officer i requires that

( ) ( ), , , , , , , , , ,

, , , , , ,

, , , , , , , , , ,

if o u o u

o u o u

C o i u i o u i r c c C o i u i o u i r c c

i r c c i r c c

V p p r c c s V p p r c c s

s s′

′

′=

= (6)

and

( ) ( ), , , , , , , , ', ,

, , , , , ,

, , , , , , , , , ,

if o u o u

o u o u

NC o i u i o u i r c c NC o i u i o u i r c c

i r c c i r c c

V p p r c c s V p p r c c s

s s ′

′=

= (7)

Notice that we do not require that , , , ,o u o ur c c r c cπ π ′= . Officers may prefer to search one race

or another because of differences in guilt probabilities; these appear in the expected

utility, equation (3), from a given group-specific search rate.

One apparently odd feature of this definition of no taste-based discrimination is

that it only imposes an equal utility requirement when equal effort is applied to both

groups. Our reasoning for this parallels the incorporation of search effort in the payoff

function. If utility were convex in effort, this would create the possibility of ruling out

taste-based discrimination against blacks if an officer spends all effort on blacks, because

he would, under the counterfactual, derive equal utility from applying all effort to whites.

We will impose a form of concavity later on, so this possibility is moot for us.5

Further, we think our definition of the absence of taste-based discrimination is

appropriate. In our view, the defining property of the absence of taste-based

discrimination is that a police officer should experience the same utility from a given

action applied to groups that are identical except for race. This definition follows the

5Ayres (2005) implicitly criticizes this definition of the absence of taste-based discrimination on another grounds, namely that the variables that comprise ,o uc c may be correlated with race and so mask disparate treatments by race. His argument may have ethical and legal merit, but it is not germane to our objective which is to understand the observable implications of race based discrimination. We return to this issue below.

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spirit of Levin and Robbins (1983) who focus on conditional exchangeability as a notion

of lack of discrimination; similar reasoning appears in Heckman and Siegelman (1993).

Search rates are simply one more variable that has to be held constant to engage in race-

based comparisons of utility. If it were the case that increasing returns implied that there

are multiple equilibrium search configurations on the part of officers, for no taste

discrimination to exist, we would require that group/search rate pairs exhibit conditional

exchangeability with respect to race. Put differently, taste-based discrimination means

that a preference for stopping blacks is the source of discrepancies in treatment, as

opposed to a feature of the search technology, for example, which implies differential

treatment for some group occurs under race neutral preferences.

One could consider empirically implementing the model we just described as a

discrete binary choice in which the officer decides whether or not to stop and search each

civilian he encounters. By doing this, one could in principle recover preferences and test

the no discrimination hypothesis directly. The main problem with the binary choice

implementation is that it requires the analyst to recover the probability that an officer

searches an individual from the civilian population as a whole. This in turns requires the

analyst to know not only the distribution of characteristics of the individuals who are

actually searched but also the distribution of characteristics of individuals who are not

searched. This latter distribution is not directly observed in data sets on police stops and

searches.6

One of the important insights in the reformulation of the problem in KPT and in

Persico (2009) is that it is possible to test an empirical implication of no discrimination

directly, sidestepping the need to recover the distribution of the unobservables. In fact,

under the assumptions of their model, in equilibrium there is no selection on

unobservables (

uc ), as guilt probabilities equalize across groups. Bjerk (2007) shows that

when selection on unobservables does occur, KPT’s finding of equal guilt probabilities,

which applies to the population, would not apply to the selected sample of potential

6This, in part, motivates different benchmarks used in studies, such as racial composition of area based on CPS data, as a means of detecting racial profiling. See Ridgeway (2007).

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criminals who are actually searched. 7

Persico (2009) uses a specific version of the officer’s expected utility calculation

In order to avoid this selection on unobservables

problem, we follow a similar approach to KPT in that we focus on implications that can

be tested by looking only at the treated sample. However, a significant difference is that

the issue of unobservables remains an important part of our analysis.

(3) in that he assumes that ( ),, , , ,, , , ,,o uC o i o u i r c c ru iV p p r c c s β= , which means that officers

only (potentially) care about the race of a guilty individual and that differences in the

weights assigned to races do not covary with either officer or other group characteristics.

He further assumes that ( ), , , ,,, , , , , 0o uNC o i o u i c cu i rV p p r c c s = so that there is no utility from

searches per se. Taste-based discrimination is defined as r rβ β ′≠ for any r r′≠ . If taste-

based discrimination is absent, officers will only search members of the race with the

highest guilt probability, rπ and effort across races is allocated to equalize guilt rates. If

the ratio of guilt probabilities between races is not equal to 1, this is evidence that the

taste for capturing criminals varies by race.

This is essentially the same logic employed in the earlier work by KPT, except

that in KPT ( ) ( ), , , , , , , , ,,, , , , , , , , , ,o u o uC o i o u i r c c Nu i C o i u i o u i r c cV p p r c c s V p p r c c s β− = and racial

prejudice enters through differential costs to discrimination. In terms of functional form

this is analogous to utility from searching someone who is not carrying contraband. They

also focus on a general equilibrium setting in which a mixed-stop equilibrium occurs, i.e.,

one where members of all groups are searched and thus the guilt probabilities are equal

across all subgroups in the absence of differential costs to search across races. KPT find

that the guilt probabilities in their traffic stop data are approximately equal across blacks

and whites. This leads them to conclude that the data are consistent with no

discrimination against blacks. As our formulation demonstrates, the KPT and Persico

(2009) analyses are based on a special case of an abstract officer decision problem. In

our subsequent analysis, we work with a particular officer decision problem of which the

KPT and Persico (2009) analyses are special cases. Although we call our decision 7Dharmapala and Ross (2004) also argue that if some subset of the population is never searched, then the result of equal guilt probabilities in KPT does not hold.

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problem general, it is still based on parametric assumptions on the officer objective

functions.

We are unaware of any work that attempts to test for taste-based discrimination in

police stops and searches without restrictions on police preferences. In the absence of

any assumptions on the objective functions of either potential criminals or the police, we

conjecture that there are no testable implications for taste-based discrimination if one

focuses on data on searches and their outcomes. A heuristic justification for this view is

the following. Suppose that ( ), , , , ,, , , , , 0o uNC o i u i o u i r c cV p p r c c s = and that

( ), , , , ,, , , , ,o uC o i u i o u i r c cV p p r c c s does not depend upon race. Suppose that data are available

on two objects. First, the average search rate for individuals with race r and observable

characteristics oc by officers with observable characteristics ,o ip :

, , , , ,, , , , , , , , ,o i o o i u i o u u i u o i op r c p p r c c p c p r cs s dF= ∫ . (8)

Second, the number of guilty individuals with race r and observable characteristics oc

who are searched,

( ),, , , , ,,, , ,o o u o u u oor c r cr c r c o u c c c r cg n r c c s s dFπ= ∫ . (9)

Since one is free to choose any distribution functions , ,, , ,u i u o i op c p r cF and ,u oc r cF and any

utility function that does not depend on race, ( ), , , , ,, , , ,o uC o i u i o u i r c cV p p c c s , it is difficult to

see how the abstract search and guilt model can place any restrictions on these data.

For example, in the spirit of AF, suppose that one tests for the absence of taste-

based discrimination by determining whether the ranking of search rates for two groups

with identical values of oc but different races, r and r′ , is preserved across two officers

,i i′ with observable characteristics ,o ip and ,o ip ′ . Clearly, one could construct a pair of

functions ( ), , , , ,, , , ,o uC o i u i o u i r c cV p p c c s and ( ), , , , ,, , , ,

o uC o i u i o u i r c cV p p c c s′ ′ ′ together with

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associated distribution functions , ,, | , , , | , ,,

u u o i o u u o i op c p r c p c p r cF F′

and | , u oc r cF to produce any

ranking across the two officers. The problem is that solely restricting

( ), ,, ,, , , , ,o uC o i u i r ci o u cV p p c c s so that it does not depend on race leaves too many degrees of

freedom in its possible forms.

This example illustrates why the literature on taste-based discrimination in racial

profiling has not proceeded nonparametrically. The example, should not, in our

judgment, be used to dismiss the KPT and Persico (2009) work, let alone the broader

literature that followed. Rather, it suggests the importance of understanding how

functional form assumptions influence the evaluation of taste-based discrimination. We

therefore study the empirical implications of taste-based discrimination in a model that

nests essential aspects of KPT and Persico (2009) as special cases.

Section 3. A generalized search and guilt model

In order to understand how the introduction of alternative preference structures

can affect claims of no discrimination while at the same time see how some relaxation of

the stringent KPT assumptions can still allow of observational implications of taste-based

discrimination, we extend KPT using a device of Persico (2009). We emphasize that an

important contribution of his paper is to set up a framework for search allocations across

groups given a time constraint. Our contribution involves relaxing aspects of his (and

KPT’s) functional form assumptions as well as some of the informational assumptions

implicitly assumed in their analyses. Relative to our abstract formulation, we assume that

( ) ( ) 1, , , , ,, , , , , , , , ,, , , , ,

o u o uu uo oi iC o i u i o u r c c c i r cr c i r c ccV p p r c c s b sαβ −= + (10)

and

( ), , , , , , , ,1

, , ,, , , , , .o u o u o uNC o i u i o u i i r c cr c c i r c cV p p r c c s b sα−= (11)

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From this formulation, one can interpret 1, , ,, , ,o u o uii r c cc r csαβ − as the (extra) utility of searching a

guilty person in distinction to 1, , ,, , ,o u o uii r c cc r cb sα− , the utility of the act of searching itself.

Since the total number of searches of members of a group by officer i is , , , , ,o u o ur c c i r c cn s , the

overall expected utility from a given allocation of searches across groups by officer i is

( ), , , , , , , , , , , ,, .o u o u o u o u o ui r c c r c c i r c r c c i rc c c u ob n s dc dc drαβ π +∫ (12)

For our specification, the term , , , uoi r c cβ corresponds to Persico’s (2009) measure of

taste-based discrimination as it characterizes how payoffs for searching the guilty depend

on the group of which an individual is a member. In addition to this potential source of

taste-based discrimination, we introduce the parameter , , , uoi r c cb in order to allow for

payoffs to depend on the act of stopping a civilian regardless of his guilt. This factor is

related to what we would argue is the most clearly unambiguous morally objectionable

aspect of profiling, the disproportionate searching of innocent blacks (Durlauf (2006)).

In order to fully understand the relationship between the searching of innocent blacks and

taste-based discrimination we need to allow for the possibility that there is utility from

their being searched, not just that they are searched as a consequence of a biased desire to

find guilty blacks.

This formulation embeds the Persico (2009) preference structure as a special case

in which 1α = ; we are also interested in cases where 1α < .8

1α <

It also embeds the KPT

preferences, with a slightly different set of assumptions on the source of taste-based

discrimination, as described in the general problem above. One value of is that it

allows for interior solutions to the individual officers’ allocations of searches across

races, as will be seen below. It is important to recognize that our formulation implies a 8Persico (2009) focuses on a setting where individuals maximize total search effort rather than search rates. While the implications for group size differ across the two models, the implications for relative search effort are similar. We prefer the search rate model because it has the property that in a no-discrimination equilibrium where individuals were simply searched at random, the search rates would be equal across races.

15

substantive preference difference relative to KPT and Persico (2009). We assume that

officers have a preference against concentrating their searches on a single group, which

renders the search rates for each officer determinate. This is not the case for Persico

(2009) or KPT in the presence of time constraints, as individual officers, in equilibrium,

are indifferent in their search allocations across groups.9

We defend our preference structure at two levels. First, from the vantage point of

functional forms, our structure nests KPT and Persico (2009) smoothly in the sense

(made precise below) that as

1α → from below, the behavior of officers in our model

converges to the appropriate analog of theirs. Hence we can investigate robustness of

their findings with respect to a class of parametric specifications.

Second, there are substantive reasons why 1α < might better describe officer

payoffs than 1α = . One reason concerns the costs of searching different groups. It may

be the case that these costs are convex in the number of searches of a given group (and

hence convex in the search rate); 1α < proxies for this possibility. Convexity in costs, in

turn, implies concavity in payoffs from group-specific search rates, as implied by 1α < .

Convexity of costs may occur if higher search rates for a given group, relative to their

population fraction, require additional actions on the part of an officer, for example by

choosing different locations from which to search.10 Convexity in costs, in turn, implies

concavity in payoffs from the search intensity. Alternatively, one can imagine that

litigation-wary police departments impose costs on officers who deviate from equal

search strategies.11

9To be precise, KPT does not constrain the number of officer stops to meet some fixed time, so

Sanga (2009) provides some evidence that the litigation affects police

behavior. He replicates the KPT analysis and extends it to include other highways in

Maryland that were not the focus of the original racial profiling lawsuit in the state,

which initiated the collection of police stops data. The finding of equal hit rates across

α would be irrelevant for their context; if they did implement a time constraint, their analysis would map to the case where 1α = . 10This would likely be the case if the groups used to determine group-specific search rates are formed based on a finely detailed set of characteristics. 11Of course, these conceptual arguments do not justify our particular functional form, which is chosen for analytical convenience and because it nests the KPT and Persico preference structures.

16

blacks and whites is not robust to the inclusion of these other highways, even after

controlling for location fixed effects. Furthermore, he finds more evidence of disparities

in White and Hispanic hit rates. We believe that a reasonable interpretation of this

finding could be that the equal hit rates found in the original KPT analysis were a direct

result of the litigation. .

An alternative view of convex costs to group-specific search is that they are

psychological, i.e. the mental effort involved in identifying groups is increasing in the

complexity of their description. Hence a random search involves less mental effort than a

search that is trying to oversample blacks; and a search that looks for violations that serve

as pretexts for traffic stops and searches (which is technically required for a stop to be

legal) requires less effort than a search that requires concentration to identify both traffic

violations and the race of the driver. This view of attention as requiring effort, with

attention on multiple attributes or objects requiring greater effort, is standard in

psychology; Kahneman (1973) is the classic reference.12

One feature of the function

A recent example of

psychological research of this type is Warm, Parasuraman, and Matthews (2008) who

review evidence on the mental energy costs of vigilance, which they defined as “the

ability of organisms to maintain their focus of attention…over periods of time.” (p. 433).

Vigilance seems an appropriate way to think about police who are observing motorists

and making decisions on who to search.

, , ,o ui r c csα , 1α < , is that it implies that a police offer

would experience lower stop costs if he were to focus on cruder group designations than

( ), ,o ur c c . This is also consistent with existing psychological evidence and in fact

provides a basis for understanding why individuals stereotype, see Macrae, Milne, and

Bodenhausen (1994). The authors specifically describe stereotypes as “energy saving

devices” (p.37) which is consistent with our functional form assumption as cruder group

descriptions are isomorphic to greater stereotyping.

That said, our objective is not to argue that one value of α is more defensible

than another, but rather to argue that there do not exist strong a priori reasons to privilege

12Cognitive costs to attention have been proposed in other economic contexts, see Banerjee and Mullainathan (2008) for an example.

17

1α = over 1α < . As suggested in the previous section, without functional form

assumptions there is little to be said about taste-based discrimination and racial profiling.

Hence it is important to understand how different functional forms affect claims of bias.

In the case of α , this functional form assumption maps to a substantive assumption on

the costs of search and hence is interpretable in terms of the underlying search

technology.

Applying the general definition of no taste-based discrimination, equations (6)

and (7), to the preferences embedded in equation (12), the absence of taste-based

discrimination on the part of officer i requires that

, , , , , , , ,o u o u o ui r c c i r c c i c cβ β β′= = (13)

and

, , , , , , , , o u o u o ui r c c i r c c i c cb b b′= = (14)

We thus distinguish between discrimination with respect to the utility of searching a

guilty driver versus an innocent driver. This general view of the way that taste-based

discrimination can enter individual decisions will of course matter in interpreting existing

tests of the no discrimination hypothesis as well as formulating new ones.

We describe optimal police behavior and the existence of equilibrium for this

model. For simplicity, officers are assumed to have a fixed number of searches T which

they allocate across groups,13

, ,, , , .o u o uc cr c i r u ocn s dc dc dr T≤∫ (15)

13Alternatively one can think of officers as having a technology to transform time τ , , ,o ui r c c

into stops say δ τ=, , , , ,, ,,o u o u o ucr i r i i rc c c c cn s . Then, if officers face a time constraint

,, , o ui r c c u odc dc dr tτ ≤∫ , the constraints may be rewritten as , ,, ,,o u o ur c ic cr c ui

o itn s dc dc dr Tδ

≤ ≡∫

which is the same as (15) except it is i specific.

18

The Lagrangian for the officer’s problem is therefore

( )( ), , , , , , , , , , , , , ,, , , , .o u o u o u o u o u o u o ui r c c r c c i r c c c i r c cr c r c uBR c i r c c Bo Rb n s n s dc dc dr Tαβ π µ µ+ − +∫ (16)

Under the assumption that individual officers are small relative to the population of

police and the groups under scrutiny, each officer takes aggregate search effort ( ,, o ur c cs ),

and hence , ,o ur c cπ , as given and independent of his own effort choice. The subscript on

the Lagrange multiplier BRµ refers to this being the multiplier for a best response

problem. The first order conditions of equation (16) are

( ) 1, , , , , , , , , , , , ,

o u o u o u o ui r c c r c c i r c c i r c c BR o ub s r c cαα β π µ−+ = ∀ . (17)

The optimal search rate of a police officer can be directly computed from the first

order conditions for our maximization problem and yields

( )

( )

11

, , , , , , , ,, , , 1

1, , , , , , , , , ,

.o u o u o u

o u

o u o u o u o u

i r c c r c c i r c ci r c c

i r c c r c c i r c c r c c ou

b Ts

b n dc dc dr

α

α

β π

β π

−

−′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′

+=

′ ′ ′+∫ (18)

Aggregating across police gives group-specific search rates,14

14Notice that we are implicitly assuming that individuals are sampled without replacement in this calculation. That is, we are assuming that adding the individual search rates generates the appropriate aggregate search rate since

, , ,

, ,, , , , ,

, , , ,

o uo u

o u o u

o u o u

i r c cr c c

r c c i r c cr c c r c c

i

i

aas s

n n= = =

∑∑

If individuals can be searched more than once, we would simply redefine the size of the group accordingly.

19

( )

( )

11

, , , , , , , ,, , , ,1

1, , , , , , , , , ,

, ,o u o u o u

o u o u

o u o u o u o u

i i

i r c c r c c i r c cc b r c c

i r c c r c c i r c c r c

r

c u o

c

b MTs dF

b n dc dc dr

α

βα

β π

β π

−

−′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′

+=

′ ′ ′+∫∫

(19)

where , | , ,i o uib r c cdFβ denotes the distribution of taste parameters across police conditional on

the characteristics of the group. Since the guilt probabilities for each group are a function

of aggregate search probabilities and the equilibrium search probabilities are determined

by search rates of the police, the equilibrium search rate is implicitly defined by

( )( )

( )( )

11

, , , , , , , ,, 1 , , ,

1, , , , , , , ,

,

, ,

, ,

,

,.

, ,

o u o u o u

o u o u

o u o u o u o

i

u

i

i r c c r c c i r c cc b r c c

i r c c r c c i r c c r c c

o uc

u o

r

uo

r c b MTs dF

r c c b n dc dc dr

c s

s

α

βα

β π

β π

−

−′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′

+=

′′ ′ ′ ′ ′+∫∫

(20)

Given the nonlinearity in the individual equilibrium stop choices, it is evident from

equation (20) that even if the mean or median officer is unbiased, this does not imply that

the presence of bias fails to affect aggregate group specific search rates.

Let 1,...,g G= index the groups that can be formed from all the combinations of

( ), ,o ur c c in the population. Assume the total number of groups, G , is finite. Equation

(20) defines a mapping of the space

( ) ( ) ( )21 1 2[0, ] [0, ] ... [0, ] ,G Gs MT n s MT n s MT nΩ = ∈ × ∈ × × ∈ (21)

into itself. This is a fixed point problem. The following proposition establishes existence

under mild conditions. We therefore state

Proposition 1. Existence of Nash equilibrium

20

Assume ( ), ,, , ,o uu cr cor c c sπ is continuous in , ,, ,[0, ]

o u o ur c rc ccs MT n∈ . Then a fixed

point exists for equation (20). Therefore, a Nash equilibrium exists in aggregate

officer stop choices across groups.

Proof: The claim is immediate from Brouwer’s fixed point theorem since G has been

assumed to be finite.

i. guilt rates and taste-based discrimination

We now consider the empirical restrictions that our generalized model places on

the equilibrium guilt rates across groups. It is evident from equation (18) that, as one

takes the limit 1α → from below, officer i ’s available searches T become entirely

concentrated on a single group, determined by the maximum value of

, , , , , , , ,o u o u o ui r c c r c c i r c cbβ π + across groups. This is the type of behavior found by KPT

when , , , o ui r c cβ β= and , , ,o ui r c c rb b= . For KPT this cannot be an equilibrium, as individuals

who are not being policed now have an incentive to carry contraband with probability 1

and those who are being policed with probability 1 should stop carrying contraband. This

forms the basis of their conclusion that aggregate policing must be distributed so that

guilt probabilities are equalized. Under the KPT or Persico (2009) preferences police

allocate relative search efforts until guilt rates are equalized, unless there are groups

whose guilt rates when they are never searched are lower than the guilt rates of every

group that is searched. We ignore this type of corner solution.

We can now analyze how our model restricts guilt probabilities across groups.

Equation (17) requires

( ) ( )1 1, , , , , , , , , , , , , , , , , , ,, , , .

o u o u o u o u o u o u o u o uii r c c r c c r c c i r c c i r c c r c ic r c c i r c cb s b sα αβ π β π− −′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′+ = + (22)

For expositional purposes and to facilitate comparison with KPT and Persico (2009), we

assume that officers have homogeneous nondiscriminatory preferences, which means that

21

, , , ,oo u ui r c c c cβ β= (23)

and

, , , ,oo u ui r c c c cb b= . (24)

We introduce this level of parameter homogeneity as it implies that in equilibrium, each

officer chooses an identical cross group distribution of search.15

Under this preference homogeneity, the first order condition for individual

officers therefore requires that the equilibrium aggregate searches obey

The average and

individual group search rates thus coincide for each group. We assume homogenous

preferences for most of this section. We return to officer heterogeneity in Subsection (iv).

( )( ) ( )( ) 1, ,

1, , , , , , , ,, ,, , , , , ,

o uo u o uo u o u oo u ou uc c o u c c c r c cr c o u c c cc r c cc crr c c s b s r c c s b sα αβ π β π−′−

′′+ = + . (25)

Equation (25) provides a way of describing how taste-based discrimination determines

equilibrium guilt rates for each group. For our model, this depends on the informational

content of race with respect to guilt, conditional on the other group characteristics. Let s

denote a fixed level of aggregate policing. Suppose that

( ) ( ), , , , , , .o u o ur c c s r c c sπ π ′= (26)

This condition means that given the group characteristics ( ),o uc c and equal policing rates

s , the probability of guilt is independent of whether the group consists of members of

race r or race r′ . In other words, race has no marginal predictive value for guilt or

innocence once one has accounted for the other elements of the officer’s information set.

15This follows from uniqueness of the optimal search solutions.

22

Combining (25) and (26), it is immediate that unique equal search rates are

applied to the groups and that the two groups have unique equilibrium guilt probabilities,

i.e., the realized group-specific guilt and search rates are

,, ., , , , , and s .o u o u o u o ur c r r cc c c c crc sπ π ′ ′= = (27)

This mimics the KPT finding which equates no discrimination with equal guilt

probabilities and thus identifies a dimension along which their test for no discrimination

is robust. Our equilibrium condition differs from KPT in that it also implies equal search

rates across groups; we return to this below.

The robustness of the equal guilt probability requirement breaks down when we

consider divisions of the population other than the groups we have defined. To fully

observe a group, it is necessary to observe the entire ( ), ,o ur c c triple. For this reason, the

racial profiling literature has focused on the idea of the hit rate for a given population

partition. The hit rate is defined as the fraction of searches of members of a partition who

are guilty. When one is working with the “true” groups as defined by ( ), ,o ur c c the hit

rate equals the guilt rate for the group. For groups that are defined only by observables,

the hit rate is influenced by differential search rates applied to subsets of the population

comprising the group as well as by the different guilt probabilities for these subsets. For

example, the hit rate for race r is the ratio of total guilty persons among those searched

of race r , ( ), ,,, ,, , ,o u o u o uo u r cr c cr c c c rn r c c s s dFπ∫

to the total number of searched people of

race r , , , ,o u o ur c cr rc cn s dF∫ . That is, the hit rate for race r , rh , is defined as

( ), ,,

,

, ,

, ,

, , ,.o u o u o u

o uo u

o u r c r c c c rc c

cr

r c c c r

r c c s s dFh

s dF

π= ∫

∫ (28)

Note that ,o uc c rdF , the conditional density of ( ),o uc c given race r , is the appropriate

conditional probability density in this calculation as the hit rate in this case asks what

23

percentage of members of race r who are searched are guilty. Notice that ,, o ur c cs is an

equilibrium condition that is determined by ( ), ,o ur c c and varies across the values of

,o uc c rdF .

Given (20), our functional forms restrict the race-level hit rate to the following

( ) ( )( )

( )( ), , ,

ˆ ˆ ˆ ˆ ˆ

11

, , , ,1

1, , ˆ ˆ ˆ ˆ ˆ ,, , , , ,,

,

ˆ

, , , ,

ˆ ˆ ˆ, ˆ, ˆ

,

,

o u o u o u o u

o o u o u o u o u

o u

u o u

c o u r c c

c c o u r c c c c r c

o u r c c c c c c

c u o

c rr

c c rr c c

r c c

r c c c d

c s r c s b MTdFh

s b n d d s dc r F

α

α

π β π

β π

−

−

+=

+

∫

∫ ∫ (29)

It is evident that, unless , ,o u o uc c r c c rdF dF ′= , it is possible that

.r rh h ′≠ (30)

Similarly, if one considers observables pairs ( ), or c and ( ), or c′ , it is possible that

, ,o or c r ch h ′≠ (31)

unless ,, u ou o c r cc r cdF dF ′= .16

This illustrates an essential restriction in the KPT framework relative to our

framework. The KPT loss functions require that all groups have equal guilt probabilities

if taste-based discrimination is absent regardless of any group characteristics. This

follows as a general equilibrium result. For instance, if police suspect that men are more

likely to carry contraband than women and therefore in the aggregate search them more,

women have a relative incentive to carry contraband compared to men. Thus, guilt rates

should equalize across observable characteristics, including race, unless police officers

derive higher utility from catching criminals of a particular race.

16The same argument applies if one conditions on equal stop rates as well as ( ), or c .

24

Put differently, KPT avoids the problem of unobservables because their loss

functions for officers have the property that race-correlated unobservables do not

differentially affect the marginal payoffs to searches. Hence in equilibrium,

( ), ,, , ,o uo u r c cr c c sπ is constant and factors out of (28). Our specification does not have

this feature. Indeed it is clear from (28) that equal hit rates across race or across

observable characteristics generally do not hold without this feature, so the link between

(29) and (31) is not an artifice of our functional forms. What matters in our specification

is that we allow preference weights to depend on group characteristics other than race and

that we allow for preferences for equal group allocations, other things equal ( 1α < in our

functional form). Hence the presence of unmeasured group characteristics matters for us

in a way that it does not for KPT.

AF and Bjerk (2007) also show that unobservables ( uc ) can matter in the context

of KPT’s loss function if these unobservables are a direct result of the choice to commit

the crime. For instance, a person who is carrying contraband may appear more anxious

when stopped by police than a person not carrying contraband. It was the choice to

commit the crime itself that produced the signal to the officer, and is outside the control

of the individual. Our results show that this problem does not depend on their

specifications of the source of the unobservables.

The finding that conditional probability dependences between unobservables and

race can render the identification of discrimination problematic is a standard problem in

discrimination analyses. Heckman and Siegelman (1993), Heckman (1998), and

Bohnholz and Heckman (2005) have pioneered this recognition by showing how

differences in higher moments in ,u oc r cdF and ,u oc r cdF ′ can produce spurious evidence of

discrimination in contexts ranging from success in loan applications to access to organ

transplants. Our findings follow this same line of reasoning. KPT also recognize this

problem. This motivates their focus on racial disparities in the productivity of searches

(hit rates) rather than the searches themselves, which may be a result of statistical

discrimination. The additional challenge that we highlight is that under more general loss

functions the hit rates also generally fail to solve the problem of unobservables.

25

There is a second dimension along which our model produces different results

from KPT and Persico (2009). Since our condition (27) also requires equal search rates,

it fails to match a key idea in KPT, namely that unequal search rates are consistent with

the absence of taste-based discrimination. Put differently, KPT is explicitly interested in

the case where race is informative about guilt probabilities at equal policing levels. We

therefore now follow KPT and consider the case where race is informative about guilt

probabilities, in the sense that

( ) ( ), , , , , , .o u o ur c c s r c c sπ π ′> (32)

In this case, unless 1α = , equal guilt probabilities require taste-based discrimination (i.e.

(23) and/or (24) do not hold). This is immediate from the equilibrium condition (22).

Furthermore in the absence of taste-based discrimination, the equilibrium search rates

implied by (22) require

,, ,, , , , , and s .o u o u o u o uc r c c c rr c r cc csπ π ′ ′> > (33)

From this vantage point, equal guilt probabilities can represent evidence in favor

of taste-based discrimination. For example, suppose that the utility of searching guilty

parties is independent of race, i.e. (23) holds. Equal guilt probabilities would necessarily

imply that , , , ,o u o ur c c r c cb b ′> if groups are perfectly observable to the econometrician, i.e.

there are no group level unobservables. Hence equal guilt rates would imply that officers

possess a differential taste for searching various races regardless of guilt. This difference

from KPT derives specifically from allowing officer utility to directly depend on the

distribution of searches across groups. This is a plausible assumption as it may reflect,

for example, returns to scale in the search process. It follows that hit rates for coarser

population subdivisions are not necessarily equal either even if taste-based discrimination

is absent.

One response to our argumentation is that KPT finding of equal hit rates is a knife

edge equilibrium under our alternative modeling assumptions. However, in light of

26

findings such as Sanga (2009), equal hit probabilities are not universally observed and

may well result from litigation, in which officers are obliged to constrain the

manifestation of their taste for discrimination. And of course, under convex search costs,

equal hit rates generally imply discrimination, hence the only question is why prejudiced

officers do not oversample certain groups beyond the levels producing equality. The

ability to point to an observable feature of their stopping strategy such as equal hit rates

as a defense is an example of why this may occur.17

Our findings may be summarized in the following two propositions.

Proposition 2. Lack of information about taste-based discrimination from equal hit

rates

Under our generalized stop model, the observation of equal hit rates between

groups characterized by ( ), or c and ( ), or c′ is compatible with either the

presence or the absence of taste-based discrimination on the part of police.

and

Proposition 3. Lack of information on taste-based discrimination from unequal hit

rates

Under our generalized stop model, the observation of unequal hit rates between

groups characterized by ( ), or c and ( ), or c′ is compatible with either the

presence or the absence of taste-based discrimination on the part of police.

Together, these propositions illustrate that without strong preference assumptions,

guilt probabilities are not informative about bias in police searches even when one allows

for additional control variables (which are by definition observable). Nothing above

17See also our discussion in section 4.

27

suggests that KPT’s or Persico’s (2009) methodological analyses are incorrect. Their

substantive conclusion that observed hit rates are equal in the absence of taste-based

discrimination holds in their model specification. Rather, our analysis illustrates which

assumptions are important for their claims to be valid.

It is worth noting that the well-known inframarginality problem is a special case

of our non-identification results in Propositions 2 and 3. The inframarginality problem

arises when officers would like to search more of a given group, but are prevented from

doing so by their time constraints. In this case, the average hit rate does not equal the

marginal hit rate, making it difficult to detect discrimination using averages.

Endogenizing criminal response, as in the case of KPT, helps alleviate the

inframarginality problem because individuals in a group that is more likely to be searched

respond by decreasing the probability of carrying contraband, which decreases the utility

of searching that group. In our context, we additionally incorporate the potential

diminishing marginal utility from searching a given group, which also helps alleviate the

inframarginality problem. Importantly, the identification problems we discuss above

extend beyond the inframarginality concern.

Some additional insight may be derived by imposing further assumptions. First,

we assume a constant elasticity (with respect to searches) functional form assumption on

group specific guilt probabilities, such that guilt probabilities do not directly depend on

race, i.e.

( ) ( ), ,,,, , , , ,o u o uco u r c o r cu cr c c s c c s γπ π −= (34)

Second, we assume that

, ,, 0, .,, ,o ui o uc crb i r c c= ∀ (35)

These assumptions allow for a closed form solution to aggregate search rates in

each group:

28

( )( )( )

( )( )( )

11

, 1,

,

,1

,,

,.o u

o u

o u o u

c o ur c

c c

cc

c u uco o

c cs MT

n dc dc cc

α γ

α γ

β π

β π′

− +

− +′ ′ ′

=′′ ′′∫

(36)

We may now draw several implications. Take two groups ( ),, o ur c c and ( ), ,o ur c c′ .

From (36), we see that aggregate policing is the same across races with the same

characteristics even if police are biased against certain characteristics other than race.

Second, hit rates ,, , ,oo uur c r c cch h ′ ′′= are equated for all groups with the same characteristics

independent of race. Third, the measured hit rates

( ), ,, , ,

,, ,,

, , ,u oo u o u

o u

o

u o

c co u r c r c c r cr c

r c c r cc

r c c s s dFh

s dF

π= ∫

∫ (37)

for races with the same observable characteristics ( )oc do not have to be equated across

r . We verify this last claim in detail.

To see why the absence of taste-based discrimination does not equalize , or ch and

, or ch ′ , first note that the numerator of , or ch , as seen in equation (37), has the closed form

solution

( )

( )( )( )( )

( )( )

1, , | ,

111

| ,1

,

,

,1

,( , )

,

,,

o u

o u

o

u o

o

u

u

o

u

o u r c c r c

c o uo u c r c

c c c

c

c

c o u u oc

c c s dF

c c MTc c d

n dc dcF

c

α γ

γ

α γ

γ

π

β ππ

β π

−

−

− +

− +′′ ′ ′

=

′ ′ ′ ′

∫

∫∫

(38)

This expression provides the first reason why measured hit rates can be race dependent:

the presence of ( , )o uc cπ in the numerator of the measured hit rate. While we have

assumed that guilt rates and preference are identical across individuals of different races

but with the same characteristics, integration against ,u oc r cdF renders (38) race dependent.

29

Similar arguments apply to ,, ,o u u or c c cc rs dF∫ , the denominator of (37), as it obeys

( )( )( )

( )( )

11

, |

,

,,

,

,, 11( ,

,

)u ou o

o u

o u

o u o u

cc

c

c o ur c c r cc r c

c o cu c u oc c

c c MTs dF dF

n dc dc

α γ

α γ

β π

β π

− +

− +′ ′′ ′

=′ ′′ ′

∫ ∫∫

(39)

The presence of ( ), ,o uc cπ and integration against ,u oc r cdF in the RHS of this expression

also can render (39) race-dependent.

ii. alternative loss functions and coordinated police activity

So far, we have considered a decentralized environment in which police officers

noncooperatively choose effort levels when utility depends on both overall searches and

searches of the guilty. This ignores the role of the police administration in influencing

police behavior. A natural objective function of the administration is to allocate police

searches so as to minimize overall crime. We call this the socially optimal allocation.18

For our environment crime minimization occurs when aggregate police search

rates are set via the optimization problem

Dominitz and Knowles (2006) also consider how a preference for crime minimization

affects the KPT test for racial profiling by studying the effort allocation of a single

representative police officer who can be interpreted as a police chief. We consider

whether the Dominitz and Knowles (2006) equilibrium search rates under coordination

are different from our generalized model of noncooperative search decisions.

18Justifications may be drawn from Persico (2002), Harcourt (2004) and Durlauf (2006) as to why a crime minimization criterion is a more appropriate policy objective than arrest maximization; these same arguments would also imply that crime minimization should trump officer utility from stops. Both Harcourt and Durlauf as well as Sklansky (2008, ch. 7) argue that there are ethical constraints that should impinge on the crime minimization solution when assessing social optima, but these are factors that lie outside of our model and so are ignored.

30

( )( )( ), ,

, , , ,, ,min , , , ,o u o o u

cu

r co ur co u c S cr c uc c r o Ss

r c c s s n dc dc dr MTπ µ µ− +∫ (40)

where the subscript on Sµ , the Lagrange multiplier for the aggregate police resource

constraint, refers to this being a social problem. We maintain the constant elasticity

crime function but allow for race-based differences in guilt, i.e. (34) is modified to

( ) ( ), , , ,, , , , ,o u o uo u r c c o u r c cr c c s r c c s γπ π −= (41)

to allow for race to have intrinsic predictive value in group-specific guilt. The first order

condition for (40), under assumption (41) requires that the marginal product of policing

between groups be equalized. This means

( ) ( )1 1, , , ,, , , , ,

o u o uo u r c c o u r c cr c c s r c c sγ γγ γπ π− − − −′ ′ ′′ ′ ′= (42)

so that the guilt rates across groups are related to search rates by

( )( )

, , , ,

, , , ,

, ,.

, ,o u o u

o u o u

o u r c c r c c

o u r c c r c c

r c c s sr c c s s

γ

γ

ππ

−

−′ ′ ′ ′ ′ ′

=′ ′ ′

(43)

We contrast this with the equilibrium guilt probabilities when officers choose

search rates noncooperatively. Suppose one assumes the Persico (2009)

nondiscriminatory preferences in the sense that , , , ,, 0o u o ur c c r c cbβ β≡ ≡ . Following (17),

the ratio of equilibrium guilt rates in the noncooperative model is

( )( )

1, , , ,

1, , , ,

, ,.

, ,o u o u

o u o u

o u r c c r c c

o u r c c r c c

r c c s sr c c s s

γ

γ

α

α

ππ

− −

− −′ ′ ′ ′ ′ ′

=′ ′ ′

(44)

As 0α → , the ratios of guilt probabilities of the social planner and noncooperative

equilibria coincide as the search rates coincide. Recall that α measures the extent to

31

which more intensive search of a group diminishes the payoffs from stops of guilty

drivers; as we have assumed that utility of stops of innocent drivers is 0. Sufficient

concavity of this payoff, i.e. sufficiently small α can render the planner and

noncooperative equilibria arbitrarily close to one another. Therefore, in the absence of

discrimination, there is no necessary difference in the implications of noncooperative and

social planning views of search rate determination.

Proposition 4. Lack of information about objective functions from hit rates

For the generalized choice model, hit rates across observed groups do not

necessarily distinguish between a coordinated distribution of searches that

minimizes crime and a noncooperative equilibrium in which police searches

generate utility based on arrest maximization and search rate intensity.

To be clear, the observational equivalence of the planner and noncooperative

equilibria that we state requires (41). But this also suggests that distinguishing between

the types of equilibria is difficult when (41) is a good approximation of the guilt rate

process and when marginal increases in search rates are extremely costly to an officer.

iii. lack of information from effort differences

We now discuss what information can be revealed on taste-based discrimination

from differences in police effort across groups without restrictions on unobservables. We

continue to assume (35) and that police officers are homogeneous (23). We also assume

that

( ) ( ), , ,,, ,, ,,o u o uo u r c o u rc ccc s c sr c r c γπ π −= (45)

holds. In analogy to (36), the unique group specific search rate is

32

( )( )( )

( )( )( )

,

11

, 11

, ,

,

,

, ,.

, ,

o u

o u

o u o u

c o ucc

o u u

r c

c c r c oc

r c c MTs

r c c n dc dc dr

α γ

α γ

β π

β π

− +

− +′ ′ ′ ′ ′

=′ ′ ′ ′′ ′∫

(46)

From (46), the equilibrium ratio of policing on group ( ), or c to group ( ), or c′ is

( )( )( )

( )( )( )

11

| ,,1

,|

,

, ,1

, ,

, ,

o u u oo

oo u u o

c o u r cr c

r cc o

c c

c cu r c

r c c dFss r c c dF

α γ

α γ

β π

β π

− +

− +′′

=′

∫

∫ (47)

when taste-based discrimination is absent. We may now state a strong observational

equivalence result.

Proposition 5. Lack of information about taste-based discrimination from

differences in search rates applied to different races

Let ,

,

o

o

r c

r c

ss ′

be any observed ratio of policing on race r and race r′ where the

observed characteristics oc are the same. Then, there exist values of

, , ,, , ,o u u o u oc c c r c c r cdF dFβ ′ ( ), ,o ur c cπ , and ( ), ,o ur c cπ ′ such that (47) holds.

As an extreme example suppose ( ), ,,o uc c o ur c cβ π and ( ), , ,

o uc uc or c cβ π ′ each take

only two values one of which is zero and the other is positive and identical for both.

Suppose the two probability densities over which integration occurs in the variant of (46)

that is germane to our assumptions only place positive mass on these two values. For this

case, the ratio of search rates obeys ( )( )

,

,

Pr ,Pr ,

o

o

r c o

r c o

s r cs r c′

=′

where the ( )Pr ,⋅ ⋅ terms are the

conditional probabilities of the positive value. Since these probabilities can be anywhere

between zero and one, we can choose them so their ratio is any positive number or zero

33

or infinity. For more general cases, it is immediately evident that we have enough

“degrees of freedom” to implement (47) when we are free to choose

, , ,, , ,o u u o u oc c c r c c r cdF dFβ ′ ( ), ,o ur c cπ , and ( )', ,o ur c cπ such that (46) holds.

While this result is mathematically trivial it is important for identification issues.

Even though the police above are not racially biased and the crime propensity functions

of the two races are exactly the same, the conditional distributions of unobservables can

be such that any observed ratio of policing across the races can be observed. Turning to

actual policy debates the main public policy concern is “excessive” policing of blacks

given the same observed characteristics as whites. As Proposition 5 demonstrates, one

cannot conclude from the observation of “excessive” policing of blacks relative to whites

that the police are biased. This of course is an essential insight of KPT; our proposition

demonstrates that their finding holds across alternative specifications.

iv. lack of information from officer comparisons

As illustrated above, the key difference between our analysis and KPT is that

unobservables affect guilt probabilities in our setting, thus making it difficult to detect

taste-based discrimination. AF introduces a role for unobservables in the KPT setting by

focusing on a particular type of unobservable that signals a potential criminal’s guilt to

the police officer. As opposed to KPT where in equilibrium guilt rates are equalized

across unobservable characteristics, this is no longer the case. In the AF setting, these

unobservables lead to differences across observed (to the econometrician) hit rates

because they are behaviors observed by the officer that are beyond the individual’s

control. Officers use this unobservable to predict the guilt of the potential criminal and

the criminal cannot eliminate the signal so that guilt probabilities do not equalize across

the unobservable in equilibrium. An example AF use is that an individual who is carrying

contraband may appear more nervous or agitated during a police stop which signals guilt

to the officer.

Given these unobservables, AF draws a conclusion parallel to our Proposition 2,

namely, equal hit rates across observed groups is no longer an implication of the absence

34

of taste-based discrimination. Notice, however, that our setting does not restrict the role

of the unobservables to that of a signal of guilt. We permit unobservables to matter in

determining police officer preferences and do not specify the distribution of

unobservables.

To obtain testable implications, AF assumes that the unobservable is single-

dimensional and that it satisfies a monotone likelihood ratio property, so that effectively

higher values of the unobservable mean that the individual is more likely to be guilty.

AF also generalizes KPT’s model by introducing officer heterogeneity, such that the cost

to searching criminals of different races varies by the race of the officer.

Under these restrictions, AF’s main empirical implication is that the relative

rankings of search rates (and resulting hit rates) are preserved across officers if the

officers are racially unbiased. In particular, the AF rank test (AF (2006, page 131, pages

145 and 146, equations (6) and (7)) in our notation is

, , , , , , , ,

, , , , , , , ,

and

and .o o o o

o o oo

i r c i r c i r c i r c

i r c i r c i r c i r c

s s h h

s s h h′ ′

′ ′ ′ ′ ′ ′

> < ⇒

> < (48)

In words, if officer i searches a potential criminal of race r at a higher rate with a lower

hit rate than officer i′ , the same must also be true in their relative behaviors toward race

r′ if one officer is not relatively more racially-biased than another. AF is particularly

concerned with heterogeneity across black, white and Hispanic officers. We allow the

heterogeneity to be individual-officer specific, but our argument could clearly be applied

to their level of aggregation. In what follows, we consider whether a similar rank

condition holds in our framework under equivalent assumptions on the unobservables.

Mapping the AF analysis into our framework, searches of a group are still

determined by the values of the triple ( ), , uor c c .19

19We do not need to take a stance on the source of unobservables. As stated above, unobservables retain meaning in our model because we assume a less restrictive functional form than KPT.

Similar to AF, we make the following

assumptions. We restrict the role of the unobservables as follows. First, we assume that

35

uc is a scalar random variable. We further restrict its role so that the main implication of

AF’s monotone likelihood assumption holds, namely that , ,o ur c cπ is monotonically

increasing in uc . Second, we mimic AF’s officers’ preferences by imposing , , , 1o ui r c cβ =

and , , , .o ui r c c ib b= if officers are unbiased, while maintaining the assumption of weakly

diminishing marginal utility from searches, i.e. 1α < .

Under these assumptions, if the relative search rates for two officers ,i i′ satisfy

( )( )

( )( )

11

| ,, ,1

,

, ,

, , ,, 1

|

1,o u u oo

oo u u o

r c c c

r c c

i r ci r c

ci r c

i r c

b dFss b dF

α

α

π

π′′

−

−

+= >

+

∫

∫ (49)

then it does not necessarily follow that

( )( )

( )( )

, ,

,

11

| ,, ,1

, , 1|, ,

1,o u u oo

oo u u o

i r ci r c

i r

r c c c

r c cc

ci r c

b dFss b dF

α

α

π

π

′ ′′

′ ′

−

−′ ′ ′

+= >

+

∫

∫ (50)

even if and , , ,o u o ur c c c cπ π= . Since the distribution of the unobservables depends

on race, condition (49) does not provide enough of a restriction to determine whether (50)

holds.

Thus, AF’s rank condition (48) does not necessarily hold even in the absence of

discrimination. This is precisely Heckman’s argument about the effects of the conditional

density of unobservables confounding tests for taste-based discrimination. The key is

that unobservables play a different role than in AF’s analysis. Suppose we start with a

situation where (49) and (50) hold. We can change the distribution of unobservables,

while maintaining , ,o ur c cπ is monotonically increasing in uc and violate the rank

condition.

36

Rank condition (48) may also fail even if we replace , , ,o u o ur c c c cπ π= with

. Only when the distributions do not depend on race ( | , |u o u oc r c c cdF dF= )

and the guilt probabilities are independent of race ( , , ,o u o ur c c c cπ π= ), does AF’s condition

necessarily follow since nothing depends on race anymore.

Section 4. Uncovering biased officers

In this section we consider the problem of a police administration (generically

referred to as the police chief) who, intent on removing biased officers from the police

force, follows two alternative commonly suggested policies.20

( ), ,o uc cr

In the first case, he

eliminates racial profiling by making sure officers’ search rates are the same across

groups that differ only by race. In the second policy he makes sure officers’ hit rates are

the same across groups that differ only by race. In particular we want to understand

whether biased officers could survive if either policy is implemented. Since throughout

we assume that the police chief has access to the same information as an officer regarding

the criminals, i.e. to , the distinction between oc and uc is not relevant. We

simplify notation by letting ,( )o uc c c= .

i. equal search rates

Suppose that a police chief implements a policy where officers who do not have

equal search rates across groups that differ only by race are fired. We first consider what

the chief would find if officers do not take the policy into account when choosing who to

search. We assume that , , 0i r cb = . From (18) it follows that the ratio of search rates for

two groups differing only by race is

20In this section, we do not explicitly prove existence of an equilibrium for the constrained problems we present. Instead, we simply ask whether whenever an equilibrium exists biased officers can adjust their behavior and still act on their bias.

37

11

, , ,

, , ,

,

, ,

, .i ri r c

i

c r c

cr i r c r c

ss

αβ πβ π

−

′ ′ ′

=

(51)

Biased officers can survive if the extent of their bias is such that this ratio equals one.

Unbiased officers would not survive unless guilt rates were equal across races.

This conclusion ignores strategic behavior on the part of prejudiced officers. If

officers take the policy into account when choosing their search rates, the question of

interest is whether biased officers adjust their behavior to equalize search rates, while still

acting on their bias. Officers choose search rates conditional on them being equal across

groups that differ only by race so they face the constraint

, , , ', , .i r c i r c i cs s s= = (52)

The officer chooses search rates , ,i r cs to maximize his utility , ,, , , ,i r c r c r c i r cn s dcdrαβ π∫

subject to both (52) and the time constraint , , ,r c i r cn s dcdr T≤∫ .

Replacing (52) into the time constraint we can rewrite it as

, , , .i c r c c i cs n drdc n s dc T= ≤∫ ∫ ∫ (53)

If we also replace (52) into the utility function, we have the modified objective function

( ), , ,, , .i r c r c r ci cs n dr dcα β π∫ ∫ (54)

Equations (53) and (54) produce the Lagrangian for the officer’s problem

( ), , , ,, , ,i r c r c r c ES c i c Ei c Ss n dr n s dc Tα β π µ µ− +∫ ∫ (55)

38

where the subscript on ESµ , the Lagrange multiplier for the officer’s time constraint

under (52), refers to this being the equal search rates problem . The solution to (55)

produces an optimal search rate given by

11

, , , ,

, , , 11

, , , ,

.

i r c r c r c

c

i r c i c

i r

c

c r c r c

n drn

s s Tn dr

dcn

α

α

α

β π

β π

−

−′ ′ ′ ′ ′ ′

′

=

≡′

′

∫

∫∫

(56)

By construction any officer, whether he is racially biased or not, can survive the policy

followed by the police chief.

To illustrate how racial bias is reflected in the differential behavior between

biased and unbiased officers, assume that tastes do not depend on the characteristics c of

the potential criminal, so that , , ,i r c i rβ β= for every officer. Assume further that officer i

is biased, i.e. , ,i r i rβ β ′> , and officer i′ is not biased, i.e., , ,i ir rβ β′ ′ ′= . Then the optimal

search rates in equation (56) are such that officer i focuses his searches more intensely

on c -groups with higher proportions of guilty criminals of race r relative to officer i′ .

As a result, the total search rate of potential criminals of race r is higher for officer i

relative to the unbiased officer.

This example illustrates that officers who equalize search rates for groups that

differ only by race can still act on their biased preferences. The margin of discrimination

in this context appears in the allocation of searches across c -groups. For example if black

males disproportionately wear baggy-pants relative to white-males, an officer who wants

to discriminate against black males in this setting searches baggy-pant-wearing males at

higher rates than non-baggy-pant-wearing males. We return to this issue in Section 5.

ii. equal hit rates

39

Suppose that the police chief now imposes the objective of equal hit rates across

groups that differ only by race. Since the hit rate and the guilt probability are the same at

the ( , )r c level, this means that officers need to satisfy

, .r c c rπ π= ∀ (57)

As before, we first consider what the police chief would find if officers do not respond to

such a policy. From our discussion in Section 3, guilt probabilities are generally not

equalized, even for unbiased officers. Hence the majority of officers would not survive

the policy.

Now, suppose that we allow officers to respond to the introduction of the policy.

In this case, the officer chooses search rates , ,i r cs to maximize his utility

, ,, , , ,i r c r c r c i r cn s dcdrαβ π∫ subject to both (56) and the time constraint , , ,r c i r cn s dcdr T≤∫ .

Replacing (56) into the utility function, we obtain ( ), , ,, ,c i r c r cc i rn s dr dcαπ β∫ ∫ . The

Lagrangian describing the officer’s problem is then

( ), , , , , , , , ,c i r c r c i r c EH r c i r c EHn s n s drdc Tαπ β µ µ− +∫ ∫ (58)

where EHµ is the Lagrange multiplier for the equal hit rates problem.

The solution to this problem is given by the optimal search rate,

( )

( )

11

, ,, , 1

1, , ,

.c i r ci r c

c i r c r c r ds T

n cd

α

α

π β

π β

−

−′ ′ ′ ′ ′ ′

=′∫

(59)

By construction the hit rates for groups that differ only by race are equal and so both

biased and unbiased officers survive this policy. Furthermore, hit rates are also equal

across individuals.

40

To understand how biased officers act on their bias in this context, take the ratio

of search rates for two groups differing only by race

11

, , , ,

, , , ,

.i r c i r c

i r c i r c

ss

αββ

−

′ ′

=

(60)

Unbiased officers also equalize search rates for groups that differ only by race. Biased

officers on the other hand, search groups proportionally to their bias. If we further look at

search rates at the race level we get

( )( )

11

, , |,1

, 1ˆ ˆ ˆ, , |

,c i r c c ri r

i rc i r c c r

dFss dF

α

α

π β

π β

−

′ −′ ′

= ∫∫

(61)

and neither biased nor unbiased officers equalize search rates (or hit rates) purely based

on race if the distribution of c depends on race.

iii. equal search and hit rates

Finally, we consider the case where the police chief imposes the objective of both equal

search rates and hit rates across groups that differ only by race. The officer now chooses

to maximize his utility , ,, , , ,i r c r c r c i r cn s dcdrαβ π∫ subject to (52), (57) and the time constraint

, , ,r c i r cn s dcdr T≤∫ . In parallel to earlier reasoning substitute (52) into the time constraint

to get (53). Substituting (52) and (57) into the utility function produces

( ), ,, , .c i r c r ci cs n dr dcα π β∫ ∫ (62)

The officer chooses search rates to maximize the Lagrangian

41

( ), , , ,, ,c i r c r c EHS c i c Ei Sc Hs n dr n s dc Tα π β µ µ− +∫ ∫ (63)

where EHSµ is the Lagrange multiplier for the equal hit and search rates problem. The

solution is

11

, , ,

, , , 11

, , ,

.

c i r c r c

c

i r c i c

c

c

i r c r c

n drn

s s Tn dr

dcn

α

α

α

π β

π β

−

−′ ′ ′ ′ ′

′

=

≡′

′

∫

∫∫

(64)

By construction both biased and unbiased officers survive the policy.

The ratio of search rates for officers i and i′ depends only on the relative

proportion of individuals of a given race in each c -group, i.e.

( )( )

11

, , ,, ,1

, , 1, , ,

.i r c r ci r c

ri cr ri c c

n drss

n dr

α

α

β

β

−

′ −′

= ∫

∫ (65)

As before, bias is reflected in higher search rates for groups containing more members of

the particular race against which the officer is biased. In contrast to Subsection (i), where

officers only equalized search rates, disparities in guilt probabilities across races do not

play a role.

These examples above illustrate that when police chiefs act to eliminate racial

bias imposing common tests used in the literature, racially biased officers can survive and

continue to act on their bias. In comparison to our results in Section 3 and 4, these results

do not depend on the existence of unobservables. For example, if police chiefs use equal

hit rates to detect bias, KPT’s test may hold even in the presence of bias because police

officers adjust their behavior in response to the policy. Notice that this is not the

consequence of unobservables since the police chief has access to the same information

42

regarding the criminal as officers. Instead, they follow because of the ability of officers to

use variables that are correlated with race in their search allocations in order to indirectly

sample different races differentially.

Section 5. Testable implications from assumptions on unobservables

In the previous sections, we illustrate the difficulties in obtaining information

about the bias of police officers. We now develop examples of testable implications

about bias in our model, based on additional assumptions on the nature of unobservable

heterogeneity. These are not exhaustive. Throughout, we assume (35), i.e. that officers

only care about arresting guilty criminals.

i. information from observable group search rates with homogeneous officers

Our first example focuses on restrictions placed on search rates for observed

groups ( ), or c . Assume that officer preferences are homogeneous in the sense of (23).

Suppose further that race has no intrinsic value in predicting guilt probabilities, so that

(34) holds. Finally, assume that unobservables are such that

( ) ,, , and are both monotonically increasing in .o uo o u c c uc c c cπ β∀ (66)

For this case, one can relate assumptions about the conditional distribution of

unobservables to the search rates across races.

Proposition 6. First order stochastic dominance of unobservables and search rates

(a) If | ,u oc r cdF first order stochastically dominates | ',u oc r cdF then the ratio of the rate at

which members of race r are searched relative to race r′ is greater than 1 even

though the observed characteristics are the same

43

, ,

', ,

,

,

1.o u

o

u

uu

o

o

r c c r c

r c c r

c

c c

s dF

s dF ′

>∫∫

(67)

(b) If | , | ',u o u oc r c c r cdF dF= , then the ratio in (66) is unity.

Proof: From (39) calculate the ratio , ,

', ,

,

,

u

u u

o

o

u o

o

r c c r c

r c c r

c

c c

s dF

s dF ′

∫∫

; note that the term

( )( ),

11

,( , )o u o uc c cc o u u oc c n dc dcα γβ π − +′ ′ ′′ ′ ′′ ′∫ cancels out. Application of first order stochastic

dominance to the remaining terms in the numerator and denominator immediately gives

(a). Part (b) follows by definition.

This proposition is of interest in understanding how assumptions about

unobservables matter in interpreting racial profiling claims. The Henry Louis Gates

affair is a prominent example of alleged racial profiling.21oc For high individuals like

Gates, policy makers might assume roughly the same conditional distributions

,u oc r cdF and ',u oc r cdF . Note that Proposition 6.b implies that hit rates should be the same

across races for Gate’s level of oc . It is this latter belief that renders differential guilt and

differential policing conditional on high levels of oc to be prima facie evidence of

discrimination by police against blacks.

What conditions could lead to a low relative hit rate ratio for blacks relative to

whites when police are not racially biased? Here is an example. Police are biased

against young males who wear baggy pants and whose belts hang far below the waistline.

Pants style is not observed by the econometrician but is observed by the officers.

Suppose that the observed oc is high, as might be the case for male college students.

Suppose further that young black college males like wearing baggy pants. Finally, 21See Harcourt (2009).

44

suppose previous studies have shown that the average observed crime rate for black male

college students is the same as for white male college students. Given this, it is

reasonable to assume that conditional guilt rates for male college students are

independent of race. Since baggy-pants-biased police like to search baggy-pants-wearing

males we get a high ratio of policing of black males relative to white males. This example

reinforces the conclusions of KPT, Persico (2009) and others that the policy problems

associated with racial bias versus other types of police bias are subtle.

ii. information from comparisons of search rates across heterogeneous officers

Instead of placing restrictions on guilt probability and the distribution of

unobservables across races, we now consider what information can be learned from

comparisons across heterogeneous officers. We begin by assuming that the officer’s

preferences do not vary over unobservable characteristics, i.e. , , , , ,o u oi r c c i r cβ β= . The

relative search rates for two officers ,i i′ satisfy

( )( )

( )( )

( )( )

( )( )

1 111 1(1

| , | ,, ,1 1

, , 1 1|

), , , , , ,, ,

, ,, , |, , , , , ,

.o o u u o o u u oo o

o oo o u u o o u u o

i r c r c c c r c c ci r c

i i r ci r c r c c c r c

r c r ci r c

r cr c rc c c

dF dFss dF dF

α

α

α α

α

π π

π

β β

β πβ′ ′′ ′ ′ ′ ′

− −−

− −

= =

∫ ∫∫ ∫

(68)

Taking the ratio of relative search rates across officers, we have

), ,

, ,

, ,

1(1

, ,

, ,

, ,

,, ,,

.

o o

o o

o o

o o

i r c

i i r c

i r c

i i r

i r c

r c

i r c

r c c

ssss

αββββ

′ ′

′ ′

′ ′ ′ ′

− =

(69)

If this ratio is greater than 1, officer i exhibits relatively greater bias against race r

(versus r′ ) than officer i′ . The hit rates provide no additional information in this case

since by definition

45

( )

( )( )

( )

1 11 1

, , , , , , | , , , , , | ,, , 1 1

1 1, , , , | , , , | ,

,o u o o u u o o u o u u o

o

o o u u o o u u o

r c c i r c r c c c r c r c c r c c c r ci r c

i r c r c c c r c r c c c r c

hdF dF

dF dF

α α

α α

π β π π π

β π π

− −

− −

== ∫ ∫

∫ ∫ (70)

and so they do not depend on officer bias.

iii. multiple officer types

We conclude our analysis by considering how one can uncover the fraction of

prejudiced officers in a population. This subsection illustrates how this might be done by

further restricting the role of unobservables. This leads to partial identification results in

the spirit of Manski. For tractability, we simplify the problem by assuming that officers

form groups based only on race and that race can take only two values: and . Since

we focus on discrimination against race we impose the normalization that

across all officers. We further simplify the problem by assuming that there are only two

types of officers: “type” I officers for whom (i.e. biased officers) and “type” II

officers for whom (i.e. unbiased officers).

Under our assumptions, the search rates for type I officers are

( )

( )

11

1111' '

I r rr

r r r r r

Ts

n n

α

αα

β π

β π π

−

−−

=+

, (71)

and

( )

11'

' 1111' '

I rr

r r r r r

Tsn n

α

αα

π

β π π

−

−−

=+

. (72)

while the search rates for unbiased (type II) officers are

46

11

1 11 1

' '

II rr

r r r r

Tsn n

α

α α

π

π π

−

− −

=+

, (73)

and

11'

' 1 11 1

' '

II rr

r r r r

Tsn n

α

α α

π

π π

−

− −

=+

. (74)

If we let denote the (unknown) fraction of type I officers in the population, the

observed search rates for each race are

( )

( )

1111

1 1 111 1 11' ' ' '

(1 ),r r rr

r r r r r r r r r

MT MTs p pn n n n

αα

α α αα

β π π

β π π π π

−−

− − −−

= + −+ +

(75)

and

( )

1 11 1' '

' 1 1 111 1 11' ' ' '

(1 ).r rr

r r r r r r r r r

MT MTs p pn n n n

α α

α α αα

π π

β π π π π

− −

− − −−

= + −+ +

(76)

Notice that, given our simplifying assumptions, the guilt probabilities are known

to the econometrician since they equal the observed hit rates. It then follows that

equations (75) and (76) give us a system of 2 equations with 3 unknowns: and .

From this system one can find identification regions for the 3 parameters. For example,

for each point in a grid of values , equations (75) and (76) imply values for and

47

. If either or , then the parameter vector is rejected. By proceeding

over all points in the grid, the identified region may be recovered.

In some situations interest may be centered on the proportion of racially biased

officers . If this is the case, a lower bound on can be obtained by considering the

extreme scenario in which is so large that type I officers focus their search exclusively

on race , so that their search rates are r

Tn

. In this case the observed search rates become

11

1 11 1

' '

(1 )rr

rr r r r

MT MTs p pn n n

α

α α

π

π π

−

− −

= + −+

(77)

and

11'

' 1 11 1

' '

(1 ).rr

r r r r

MTs pn n

α

α α

π

π π

−

− −

= −+

(78)

Adding (77) and (78) and rearranging terms.

'1

1r r

r

s sp MTn

− −=

− (79)

Equation (79) provides a lower bound on since it is derived assuming the largest

possible search rate for members of race by the biased officers. Clearly the bound is

not sharp since the assumption that prejudiced officers search at rate r

Tn

may not be

consistent with the data. One easy way to sharpen the bound is to replace r

Tn

with the

highest search rate in the sample of officers. We leave more sophisticated ways of

48

sharpening the bound, which rely on the empirical distribution of search rates to future

work.

Section 6. Summary and conclusions

In this paper, we explore how various facets of the stop and guilt data can shed

light on taste-based discrimination in racial profiling. Working with parametric models

that nest some of those most prominent in the literature, we argue that mapping claims

about the absence of taste-based discrimination to relationships between searches and

guilt is sensitive to functional form assumptions and to assumptions about the nature of

unobserved group level characteristics.

Importantly, nothing we have argued suggests that KPT’s, Persico (2009)’s or

AF’s methodological analyses are incorrect, nor is our discussion intended to diminish

the value of their studies. Rather, our analysis clarifies how apparently innocuous

assumptions in these models can lead to vastly different claims about the presence or

absence of taste-based discrimination. In particular, we highlight how the general

presence of unobservables affects the robustness of their tests. Heckman (2000, 2005)

emphasizes that properties such as identification are characteristics of models and

therefore are always reliant on assumptions. We demonstrate how this is true in the racial

profiling context.

Overall, our results may appear to be discouraging for the potential to detect

discrimination in data on police stops and searches. One way forward is to start with the

KPT and Persico (2009) assumptions and generate a more general model that relaxes

some of these assumptions, while still maintaining testable implications on the presence

or absence of taste-based discrimination. Our paper provides some examples of how this

may be done.

A systematic development of models of that span the plausible assumptions that

distinguish different models of police stop and search process would produce a model

space. A researcher may then evaluate evidence of taste-based discrimination given the

model space, as opposed to the standard procedure, which amounts to evaluating

49

evidence based on a single model. Model averaging methods, for example, provide a way

to evaluate the evidence from these models. 22

A different route is to reformulate tests of taste-based discrimination in a partial

identification context; we give an example in Section 5.iii on how this may be done, but

the general idea has yet to be systematically explored. A partial identification approach

will allow for weaker assumptions on the model, particularly on unobservable

heterogeneity. Brock and Durlauf (2007) show this is possible for social interactions.

Theoretical work on partial identification in abstract profiling environments has been

done by Brock (2006) and Manski (2006a), which is an appropriate conclusion to this

paper.

As our results indicate, this will probably

require that each of the alternative models place strong assumptions on the nature and

effects of the unobserved heterogeneity that distinguishes groups observed by an

econometrician versus an officer. But the model averaging approach provides a strategy

for developing robust evidence of discrimination or its absence, i.e. evidence that is not

contingent on particular sets of assumptions.

22See Brock, Durlauf and West (2003) for a conceptual defense of model averaging and Cohen-Cole, Durlauf, Fagan, and Nagin (2009) for an example in criminology.

50

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